Let the random variable $X$ have the normal distribution $N(\mu ,{\sigma }^2)$. Explicit formulas for maximum likelihood estimator of $\mu ,\sigma$ are derived under the hypotheses $\mu +c\sigma \le m+\delta ,\mu -c\sigma \ge m-\delta$, where $c,m,\delta$ are arbitrary fixed numbers. Asymptotic distribution of the likelihood ratio statistic for testing this hypothesis is derived and some of its quantiles are presented.

In the paper a test of the hypothesis $\mu +c\sigma \le M$, $\mu -c\sigma \ge m$ on parameters of the normal distribution is presented, and explicit formulas for critical regions are derived for finite sample sizes. Asymptotic null distribution of the test statistic is investigated under the assumption, that the true distribution possesses the fourth moment.

The rank statistic $H$ based on the number of exceeding observations in two samples is suitable for testing difference in location of two samples. This paper contains tables of one-sides significance levels $P\{H\ge k\}$ for $k=7,8,...,11;max(2,n-10)<m\le n\le 25,k=9,10,...,13;max(2,n-15)<m\le n-10;13\le n\le 25;k=11,12,...,15;2<m\le n-15,18\le n\le 25$, which includes almost all practically used significance levels for $3\le m\le n\le 25$, where $m,n$ are the sample sizes.

Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(a+ba\right)$ voting sequences are equally probable. Denote by ${\alpha }_r$ and by ${\beta }_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1,\cdots ,a+b$. The purpose of this paper is to derive, for $a\ge b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1,\cdots ,a+b$ for which (i) ${\alpha }_r={\beta }_r-c$, (ii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$, (iii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c\pm 1$, where $c=0,\pm 1,\pm 2,\cdots$.

The two-sample Lepage test, devised for testing equality of the location and scale parameters against the alternative that at least for one of the parameters the equality does not hold, is extended to the general case of $k>1$ sampled populations. It is shown that its limiting distribution is the chi-square distribution with $2(k-1)$ degrees of freedom. This $k$-sample statistic is shown to yield consistent test and a formula for its noncentrality parameter under Pitman alternatives is derived. For...